1. Introduction. A variety of methods1 have been proposed to obtain ... scripts 1 and 2 represent the values for aerosols and air .... { (r') (a W)6(rI) - 1)- a9) "r') - s2. 2.
Error caused by using a constant extinction/backscattering ratio in the lidar solution Yasuhiro Sasano, Edward V. Browell, and Syed Ismail The Bernoulli solution of the lidar equation with the assumption of a constant extinction/backscattering ratio can lead to errors in the derived aerosolextinction and backscattering profiles. This paper presents a general theoretical analysis of the errors that result from differences between the assumed and actual extinction/ backscattering ratio profiles. Examples of the influence of the constant extinction/backscattering ratio assumption on the lidar derived aerosol extinction profile are presented for various laser wavelengths. 1.
Introduction
A variety of methods1 have been proposed to obtain quantitative profiles of extinction and backscattering coefficients from a Mie lidar signal.
S1 (R) aaj(R)3 #(R), 1 S2
Recently Klett 2
has proposed a general solution for the lidar equation which is applicable to cases with a variable extinction/ backscattering ratio as a function of range. "This paper presents a general theoretical analysis of the errors that result from differences between the assumed and actual extinction/backscattering ratio profiles." The error caused by using a constant extinction/backscattering ratio instead of a spatially variable one is discussed using analytical expressions and by numerical simulations at the lidar wavelengths of 300, 600, and 1064 nm.
X(R)
2(R)1 2(R).
X(R) =
[ al(R) +Sj(R) a2(R) S2
X exp {-2
J
= C[ 1(R) + 32(R)]exp {-2
We introduce the normalized total extinction coefficient y(R) defined as
J[ )
(1)
where R is the range from the lidar, P(R) is the received power, C is the system constant including the laser output power, and 3 and a are the volume backscattering and extinction coefficients, respectively. Subscripts 1 and 2 represent the values for aerosols and air molecules, respectively. The aerosol and molecular extinction/backscattering ratios are, respectively;
a2(R),
S2
and substituting it for Eq. (4), we obtain S1 (R)X(R) exp{-2
[al(r) + a2(r)]dr}
(4)
[al(r) + a2 (r)]dr} -
y(R)--a1(R)+ P(R)R2
(3)
Note that S2 is not dependent on range because of the constant relationship between a and : for molecules. From Eqs. (1)-(3) we obtain
11. Theory
Let X(R) be the lidar signal corrected for rangesquared dependence. Then X(R) can be written as
(2)
-1]
a 2(r)dr = y(R) exp [-2 J y(r)dr].
(5)
The left-hand side of Eq. (5) consists of all known quantities provided that S 1 (R) is given and 2(R) is determined by an appropriate meteorological measurement or by model atmosphere data. Taking the logarithms of both sides of Eq. (5) and differentiating them with R give d n (iSl(R)X(R)] exp {-2 JR [Sl(r)
] a2 (r)dr})
dR
Yasuhiro Sasano is with Old Dominion University Research Foundation, P.O. Box 6369, Norfolk, Virginia 23508; E. V. Browell is with
NASA Langley Research Center, Hampton, Virginia 23665; and S. Ismail is with SASC Technologies, Inc., Hampton, Virginia 23665. Received 26 June 1985.
1
dy(R)
y(R)
dR
2y(R).
(6)
This is the same expression as Eq. (7) in Klett's original paper3 except that the left-hand side is modified by a variable S1 (R) and the exponential term. Finally we solve Eq. (6) known as the Bernoulli equation to get 15 November 1985 / Vol. 24, No. 22 / APPLIED OPTICS
3929
S1(R)X(R) exp {2 JR [S)
+S1 (R)
al(R) +
-
1] a 2(r)dr
a2 (R) =
R
(RS(Ro)
L
al(Ro) + 82
JR SI
1] 2(r')dr dr
RoL82
where we put a boundary condition for the differential Eq. (6) as y(Ro) = al(Ro) + S
a2 (Ro).
82
A solution in terms of the total backscattering coefficient can be written as X(R) exp {-2 1(R)
+ 0 2(R) =
JIo
[Sl(r) - S2 112(r)dr} (8)
X(Ro) -2 #3(Ro)+ #2(Ro)
Sl(r)X(r) exp -2 JRO
[Sl(r')- S2]2(r)dr dr
ROI
Equations (7) and (8) have similar forms to those given by Fernald,4 except that the extinction/backscattering ratio is treated as a function of the range and is included in the integral calculations. The numerical calculation scheme of Fernald4 has to be modified to incorporate the differences found in Eqs. (7) and (8). The exponential term is modified as A(I,I
-
1) =
[S1(I- 1) - S212(1 + [S(I
-
-
1)
S2 1 2(IAR.
(9)
The normalized total extinction coefficient at range R(I - 1) becomes, in the backward integration mode, l(I- 1) + S
Sj(I - 1)X(I - 1) exp[A(II - 1)]
-,(I--11) = 82
S(1)X()
(10)
+ S1(I)X(I)+ Sj(I - 1)X(I - 1) exp[A(I,I - 1)]AR
S2
and the total backscattering coefficient is X(I - 1) exp[A(II - 1)]
01(I-1) + 02(1-1) =
X( ) 01)
+ 12U) + S,(h)X(I) + S,(I -
1)X(I - 1) exp[A(II - 1)]JAR
Similar expressions for the forward integration mode can be derived. The first term in the denominator of the right-hand side represents a boundary condition term in each computational step. To see the effects of neglecting the spatial variability of the extinction/backscattering ratio in the lidar signal processing, let us examine the errors in general form by extending the procedures of Klett3 and Braun5 and including the molecular component. We designate the true values by , and we obtain from Eq. (5) S1(R)X(R) exp{
J[ )-]
&2(r)dr
valueX, and also cx2may have discrepancies from the true a2 due to measurement errors or discrepancy of the model atmosphere from the real one. Defining a Si/S1, - XIX, q = y/y, and r3 a2/a2, Eqs. (5) and (11) give a(R)t(R) exp (-2
{[a(r)(r) -
-
1] S a2(r)
[6(r)- 1]a2(r)}dr)
= in(R)exp{-2
J
[r(r)- 1](r)dr} -
(12)
Taking the logarithm and differentiating the above equation give
=CS'(R)
exp[1-2Jf, (r)dr1.-
(11)
dR-_2y =-2y + d (ai) - 2(ab-1) S 2 + 2(3-1)&2,
-qdR
Here we designate X and 2 because X may have measurement errors and be different from the true 3930
APPLIED OPTICS / Vol. 24, No. 22 / 15 November 1985
dr
82
(13)
which is similar to Eq. (6) and can be easily solved to give the following expression:
n(R) =
[ exp[ -2f
+
r
(5(r)6(r)-1)
2
2(r)
-
(6(r)-i)
a
2
(r)
,
drj
a(R)
(R)/(
(Ro)M(Ro))
Ro (1 4)
~~R
r 1/n
(RO)
-2f
I
y(r)
xp
r {
-2f
Ro
(r')
- 1)a9)
(a W)6(rI)
"r')
The role of S1 (R) can be examined for cases of small optical thickness by neglecting the integral terms and simplifying Eq. (14) to 7(R) = (Ro)
-
2
s2
Ro
(6(r')-1) a2(r' )
dr']
(r)E(r)/(.(R.)C(Ro))
dr I
' E
IL U] 0 z
(R) a(Ro)
0
or
U
x
al(R) + S 82
z 0
a2 (R)
0 U]1
-1R +9(R &,a(R) +S
2lR
0
al(Ro)+ S
~o(Ro) + 8
(Ro)
1 &,(Ro) + 1 S8
with the assumption that
a2 (Ro)
L-(R) / S (Ro)1
(15)
(R) = 1 for all R terms.
a1 (Ro) [S1 (R)
al(Ro) L1S
S1j(Ro)1
/ Si(Ro).
(16)
The aerosol extinction coefficient atR is proportional to Si(R)/Si(Ro).[Sl(R)/Il(Ro)]-l. If we put a constant value for S1 (R) S 1(Ro) in spite of the variable S 1 (R), the solution for al(R) is inversely proportional to S1 (R). Taking the attenuation term into account properly [Eq. (14)], the extinction profile tends to converge to the true value as the optical thickness becomes large (when the backward integration is applied). Even in this case the profile is still affected by the change in the extinction/backscatter
ratio,
(R)/la(Ro), as is clearly
shown by Eq. (14). When the attenuation is negligible, the aerosol backscattering coefficient is independent of S1 (R). The change in a(R)/af(Ro) affects the backscattering profiles in optically thick cases. Ill.
2000
1500
Fig. 1.
Reconstructed
profiles of aerosol extinction
coefficient.
Boundary conditions were correctly given at 1500 m, and the backward integration was applied.
1
ligible, we obtain -R)
1000 RANGE ()
Assuming that the molecular extinction terms are nega1 (R)
500
c 2(Ro)
Examples and Discussion
Figure 1 depicts the results of model simulation to show the effects of neglecting the change in the extinction/backscattering ratio. The aerosol distribution was modeled as having a stepwise shape and is shown by a solid curve. The extinction/backscattering ratio p1 (R) takes the value of 40 and 80 as indicated in the figure. The curves with symbols are the aerosol extinction profiles solved for the lidar signals constructed according to the lidar equation (1) from the model
aerosol distribution, and a homogeneous molecular distribution. The received signal is assumed to have no measurement error, and the molecular distribution is assumed to be accurately known. The wavelengths used for these calculations are 300, 600, and 1064 nm, and the corresponding molecular extinction coefficients are 0.145 X 10-3 m-1 , 0.815 X 10-5 m-1 , and 0.807 X 10-6 m- 1 , respectively.
The curves with open
symbols in Fig. 1 are the results for a constant extinction/backscattering ratio S = 60, and the other two curves are for Si = 40 and 80. In each case the backward integration made is used, and the boundary conditions are specified at R = 1500 m. According to Eq. (15), when a constant value is assumed for S, the solution is affected only by the molecular term a 2(R). The difference among the three curves for 300 nm with S 1 = 40, 60, and 80 arises from
the difference in the magnitude of the molecular term. Since the molecular extinction coefficient is negligibly small compared with that of aerosols for 1064 nm, the calculated extinction coefficient becomes half of the model value at R = 1000 m, where S 1 (R) is doubled in
accordance with the simplified discussion above [see Eq. (16)]. The difference among the three curves for 300, 600, and 1064 nm with a constant S(= 60) is due
to the difference in the relative magnitude of the molecular term to the aerosol extinction coefficient. AtR = 500 m, the modeled extinction/backscattered ratio S1 (R) changes again to -40. The aerosol extinction coefficients calculated do not return immediately to the modeled values but take higher values, although the behavior here is dependent on the model of aerosol 15 November 1985 / Vol. 24, No. 22 / APPLIED OPTICS
3931
distribution including the width of the region with a high extinction coefficient. If the optical thickness is sufficiently large, the solution converges to the true value. The rate of convergence is dependent on the magnitudes of aerosol and molecular extinction coeffi-
Yasuhiro Sasano is a visiting research scientist from National Institute for Environmental Studies, Tsukuba, Ibaraki 305 Japan.
cients and also on S1.
Under most circumstances, it is not possible to know a priori the extinction/backscattering ratio as a function of the range. However, the solutions given in Eqs. (7) and (8) become realistic when information on the
extinction/backscattering ratio is available through a 6
parametrization such as given by Salemink et al. for cases which are dependent on relative humidity. Rel-
ative humidity is measured by conventional meteorological instruments and will possibly be available from DIAL water vapor and temperature measurements. By taking the spatially variable extinction/backscattering ratio into account, more accurate information
References 1. R. T. H. Collis and P. B. Russell, "Lidar Measurement of Particles
and Gases by Elastic Backscattering and Differential Absorption," in Laser Monitoring of the Atmosphere, E. D. Hinkley, Ed. (Springer-Verlag, New York, 1976), p. 117.
2. J. D. Klett, "Lidar Inversion with Variable/Extinction Ratios," Appl. Opt. 24, 1638 (1985).
3. J. D. Klett, "Stable Analytical Inversion Solution for Processing Lidar Returns," Appl. Opt. 20, 211 (1981). 4. F. G. Fernald, "Analysis of Atmospheric Lidar Observations: Some Comments," Appl. Opt. 23, 652 (1984).
can be ob-
tained.
5. C. Braun, "General Formula for the Errors in Aerosol Properties Determined from Lidar Measurements at a Single Wavelength,"
Portions of this research were supported by NASA Cooperative Agreement NCC1-28 and NASA contract
6. H. Salemink, P. Schotanus, and J. B. Bergwerff, "Quantitative Lidar at 532 nm for Vertical Extinction Profiles and the Effect of
on profiles of aerosol optical properties
Appl. Opt. 24, 925 (1985).
Relative Humidity," Appl. Phys. B34, 187 (1984).
NAS1-16115.
Steven K. Case of the University of Minnesota at the 1984 OSA
Annual Meeting. Photo: F. S. Harris, Jr.
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APPLIED OPTICS / Vol. 24, No. 22 / 15 November 1985
